CBSE Class 10 Mathematics Perimeter and Area of Plane Figures Notes Set 10

Perimeter and Area of Triangles

Perimeter is the length of the boundary and area is the surface enclosed. For a triangle with base b and height h the area is \( \frac12 bh \). With sides a, b, c and semi-perimeter \( s = \frac{a+b+c}{2} \), Heron's formula gives area \( = \sqrt{s(s-a)(s-b)(s-c)} \).
An equilateral triangle of side a has area \( \frac{\sqrt3}{4}a^2 \) and height \( \frac{\sqrt3}{2}a \). An isosceles right triangle with equal sides a has hypotenuse \( a\sqrt2 \), area \( \frac{a^2}{2} \) and perimeter \( (2+\sqrt2)a \). The circumradius of a triangle is \( R = \frac{abc}{4K} \) and the inradius is \( r = \frac Ks \) where K is the area.

Quadrilaterals

Rectangle: area lb, perimeter 2(l + b), diagonal \( \sqrt{l^2+b^2} \). Square of side a: area \( a^2 \), diagonal \( a\sqrt2 \). Parallelogram: base × height. Rhombus: \( \frac12 d_1d_2 \) and perimeter 4a. Trapezium: \( \frac12(a+b)h \) where a and b are the parallel sides.

Circle, Sector and Ring

Circle of radius r: circumference \( 2\pi r \), area \( \pi r^2 \). A sector of angle θ degrees has arc \( \frac{\theta}{360}\times2\pi r \) and area \( \frac{\theta}{360}\times\pi r^2 \). A ring between radii R and r has area \( \pi(R^2-r^2) \). A regular hexagon inscribed in a circle of radius r has side r and area \( \frac{3\sqrt3}{2}r^2 \).

Useful Ideas

When a wire is bent from one shape into another, the length of the wire (the perimeter) stays constant. For similar figures, the ratio of areas is the square of the ratio of corresponding lengths. A regular hexagon of greatest area in an equilateral triangle of side 3a has side a and area two-thirds of the triangle.

Solved Examples

Example 1. The sides of a triangle are 13 cm, 14 cm and 15 cm. Find its area and the length of the altitude on the longest side.
Answer:
s = 21, so area \( = \sqrt{21\cdot8\cdot7\cdot6} = 84 \) cm².
Altitude on 15 cm \( = \frac{2\times84}{15} = 11.2 \) cm.
84 cm² and 11.2 cm

Example 2. A rectangle has perimeter 34 cm and diagonal 13 cm. Find its area.
Answer:
\( l+b = 17 \) and \( l^2+b^2 = 169 \), so \( 2lb = 289-169 = 120 \).
Area = 60 cm²

Example 3. The diagonals of a rhombus are 24 cm and 10 cm. Find its perimeter and area.
Answer:
Side \( = \sqrt{12^2+5^2} = 13 \) cm.
Perimeter = 52 cm, area = 120 cm²

Example 4. The parallel sides of a trapezium are 20 cm and 14 cm and the distance between them is 9 cm. Find the area.
Answer:
Area \( = \frac12(20+14)\times9 = 153 \).
153 cm²

Example 5. Find the area of a sector of angle 60° in a circle of radius 6 cm (use \( \pi = 3.14 \)).
Answer:
Area \( = \frac{60}{360}\times3.14\times36 = 18.84 \).
18.84 cm²

Example 6. A rope of length 88 cm is bent first into a circle and then into a square. Find the difference between the areas (use \( \pi = \frac{22}{7} \)).
Answer:
Circle: \( 2\pi r = 88 \Rightarrow r = 14 \), area \( = 616 \). Square side 22, area 484.
616 − 484 = 132 cm²

Example 7. An equilateral triangle has side 12 cm. Find the area of its incircle (use \( \pi = 3.14 \)).
Answer:
Inradius \( = \frac{12}{2\sqrt3} = 2\sqrt3 \), so \( r^2 = 12 \).
Area \( = 3.14\times12 = 37.68 \).
37.68 cm²

Example 8. A square of side 10 cm has a circle inscribed in it. Find the area of the part of the square outside the circle (use \( \pi = 3.14 \)).
Answer:
Circle radius 5, area \( = 78.5 \).
100 − 78.5 = 21.5 cm²